Irreducibility of a general polynomial in a finite field

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Homework Help Overview

The discussion revolves around proving the irreducibility of the polynomial q(x) = x^p - x + a over the finite field F_p, where p is a prime and a is a nonzero element of F_p.

Discussion Character

  • Exploratory, Assumption checking

Approaches and Questions Raised

  • The original poster attempts to show that none of the elements of F_p are roots of the polynomial and explores the implications of a root α leading to α + 1 also being a root. There is uncertainty regarding the degree of the extension field F_p(α) and how it relates to the irreducibility of q(x). Another participant suggests examining the possibility of q(x) splitting into linear factors if it is reducible.

Discussion Status

The discussion includes attempts to connect previous work with new suggestions, indicating a collaborative exploration of the problem. The original poster expresses uncertainty about the next steps but acknowledges assistance from others. There is a mention of a resolution by the original poster, but the details of that resolution are not provided.

Contextual Notes

There is a lack of clarity regarding the degree of the extension field and the implications of the polynomial's structure on its irreducibility. The original poster's work is based on hints from a textbook, which may impose certain constraints on the approach taken.

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Homework Statement



For prime p, nonzero a \in \bold{F}_p, prove that q(x) = x^p - x + a is irreducible over \bold{F}_p.


Homework Equations





The Attempt at a Solution



It's pretty clear that none of the elements of \bold{F}_p are roots of this polynomial. Anyway, so far, following a hint from the book, I've shown that if \alpha is a root of q(x), then so is \alpha + 1, and from there I was able to deduce (hopefully not incorrectly) that \bold{F}_{p}(\alpha) is the splitting field for q(x) over the field, but I can't figure out the degree of the extension, so I'm kind of stuck. Any ideas?
 
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How about showing that if q(x) is reducible, then it must split into linear factors?
 
I sat and thought about it for about 45 minutes and played around on a chalkboard with it, and I'm not sure how to go about your suggestion. Is it related to the work I already did, or is it a completely new line of attack? Thanks for helping.
 
Okay, I figured it out. Thanks a lot.
 

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